Distributing Graph States Over Arbitrary Quantum Networks
Under construction Meignant, Markham and Grosshans (2019)
Tags:Specific Task
Assumptions
- Perfect distribution of Bell pairs occurring at perfectly synchronized times.
- Perfect node local computation.
- Ignore the cost of classical communications.
- Ignore processing time of local quantum processor.
- Ignore the cost of quantum memory.
Outline
The protocol 1 aims to distribute multipartite entangled states that are represented by graph states over fixed networks of arbitrary topology. They first introduce a protocol to distribute GHZ states that considering the assumptions takes a single time step and is optimal in terms of the Bell pair used. Their second protocol is a generalization of the first one and can distribute any arbitrary graph state using at most twice as many Bell pairs and steps than the optimal distributing protocol for the worst case scenario.
In this protocol a quantum network is represented as a graph upload image here
The physical distribution of graph states are represented as graph operations ignoring local corrections.
To distribute a GHZ states over all the nodes of an arbitrary set of the network nodes we have two steps:
- Find a minimal tree covering all the nodes of using the Steiner Tree Problem.
- Minimal tree is a subgraph connecting all the nodes in with the minimum number of edges.
- Apply an operation called Star Expansion on the leaves of the tree from the previous step.
To distribute an Arbitrary Graph State we realize multiple iterations of the protocol above.
After that we make measurements on the participating nodes to generate the arbitrary graph state we want.
Requirements
- Network Stage: Quantum Memory
- Nodal Clifford Operations.
Notation
- qubit in the center of the star graph.
- Node of the network which contains a qubit (neighborhood of b).
- number of nodes of A.
- , Represents each of the qubits of that constitutes a Bell pair with a qubit in another node of the network.
Protocol Description
Graph Operations:
- Vertex Deletion. Removes one vertex and all the associated edges from the graph.
- Implemented by the Pauli Measurement of the relevant qubit in the Z basis.
- Local Complementation. Inverts the subgraph induced by the neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_a}
of the concerned vertex .
- Implemented by the quantum operator acting on the qubits .
- Edge Addition. Add an edge between two nonadjacent vertices.
- Implemented by applying a controlled-Z between the desired nodes.
- Edge Deletion. Delete an edge between two adjacent vertices.
- Implemented by applying a controlled-Z between the desired nodes.
- Entanglement Swapping.
- Implemented by applying controlled-Z gate followed by two single qubit Y-measurement.
GHZ State Distribution Protocol
Input
- N-GHZ state: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\text{N-GHZ}\rangle = (|0\rangle^{\otimes N}+|1\rangle^{\otimes N})/\sqrt{2}} .
- That is locally equivalent to: that is called a star graph link to star graph
- Arbitrary set of the network nodes.
- Assuming that the network topology is already given to us.
Output
- N-GHZ state distributed over .
GHZ-Distribution Algorithm
- Find a minimal tree covering all the nodes of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
- is a random leaf from the tree of step 1.
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=1}
to :
- Let be any unprocessed node of such as .
- Apply the Start Expansion Algorithm on node Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} as the center of the star.
Start Expansion Algorithm This routine uses the Bell pairs of the node to add the edges to the graph state, as well as the edge iff .
- All the qubits of are linked using Controlled-Z operations between all possible pairs.
- Local complementation is applied to the qubit linked to .
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \notin W}
:
- Remove and all the edges within by -measuring it
- Else (when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in W}
):
- Keep and apply Controlled-Z gates to remove all edges within .
- Apply a -measurement on all the other qubits , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i>0} creating the desired Start Graph.
Arbitrary Graph State Distribution Protocol
To distribute an arbitrary graph state we first distribute the edge-decorated complete graph state. From this graph we can construct any other graph state by measuring each edge-qubit with a:
- Z measurement to delete this vertex and its edges
or a
- Y measurement to delete the vertex but keep the edges.
Input
- Arbitrary graph state to distribute
- Arbitrary set of the participating nodes.
Output
- Arbitrary graph state distributed over .
Arbitrary Graph State Distribution Algorithm
- Solve the Steiner Tree Problem on the network for the nodes.
- Each one of the nodes of this tree has an central leaf , .
- For to :
- Distribute a (k-1-j)-GHZ state on the node of using the GHZ-Distribution Algorithm .
- Delete vertices from the tree in order to have the covering tree for the set .
- For each one of the nodes apply local operations to generate the edge-decorated graph state.
- For each one of the nodes construct the desired arbitrary state by applying and measurements.
Further Information
The distribution of multipartite entangled states over quantum networks has also been studied in the following articles:
- Epping et all (2016) Investigate the creation of a graph state presenting the shape of the network in the presence of noise.
- Cuquet and Calsamiglia (2012) and Matsuzaki et al (2010) Present decomposition of graph states into various building blocks that can be purified and merged to construct graph states over a network.
- Hahn et all (2019) Studies the possible transformations of an already shared graph-state, with a single-qubit per location.
- Pirker and Dür (2019) Includes a protocol very similar to the one presented in this page, but the modeling of the network is different, as well as the optimized metrics. They describe a hierarchical network stack and use it to provide robustness against router or sub-network failures, which is not presented in this work.